Riemannian Geometry: Is It a 2nd Year Grad Course?

[JasonJo]

Hey folks,

Is it generally true that for US Math PhD programs, Riemannian geometry is a 2nd year grad course? I was looking at JHU, one of the PhD programs I will be applying to, and they don't require you take any prereq grad courses. And the cirriculum seems to be the standard RG cirriculum: metrics, Jacobi fields, curvature, sectional curvature, Hopf-Rinow, etc.

Some schools require that your first year of grad school is essentially Algebra I, II, Analysis I, II and Topology (with a mix of Geometry via differential forms, deRham cohomology, differentiable manifold, vector fields, tensors, lie derivatives, and so on) I and II.

What are your experiences with this? I don't see why it would be a first year course as not all students are going to be pursuing RG in graduate school anyway while all grad students should know basic grad level analysis, algebra and topology/geometry.

 

[TonyG]

Hi there,

In general, it is true that Riemannian geometry is typically a second year graduate course in US Math PhD programs. However, it can vary from school to school and some may require it to be taken in the first year. It is also important to note that while some schools may not have a specific prerequisite for Riemannian geometry, it is still expected that students have a strong background in analysis, algebra, and topology/geometry before taking the course.

In my experience, Riemannian geometry is a crucial course for any graduate student in mathematics, regardless of their specific research interests. It provides a deep understanding of the underlying structure of spaces and is often used as a tool in various areas of mathematics. Therefore, I would highly recommend taking it during your graduate studies, even if it is not a required course for your program. It will only enhance your mathematical knowledge and skills.

I hope this helps and best of luck with your PhD applications!

 

[Mene]

As a scientist familiar with Riemannian geometry, I can confirm that it is typically considered a second year graduate course in US Math PhD programs. This is because it requires a strong foundation in undergraduate mathematics, including analysis, algebra, and topology/geometry, before delving into the more advanced concepts of Riemannian geometry. Additionally, many students may not have a specific interest in RG and may choose to focus on other areas of mathematics in their first year of graduate school. However, it is important to note that each program may have different requirements and it is always best to consult with the specific program you are interested in to understand their expectations and curriculum. Ultimately, it is important for all graduate students to have a solid understanding of basic graduate level mathematics before moving on to more specialized topics like Riemannian geometry.